A359432 - OEIS

%I #12 Jan 26 2023 04:13:07

%S 1,-1,-1,-1,-1,1,-1,1,0,1,-1,1,-1,1,1,1,-1,0,-1,1,1,1,-1,-1,0,1,-2,1,

%T -1,-1,-1,-1,1,1,1,0,-1,1,1,-1,-1,-1,-1,1,0,1,-1,-1,0,0,1,1,-1,2,1,-1,

%U 1,1,-1,-1,-1,1,0,-1,1,-1,-1,1,1,-1,-1,0,-1,1,0,1,1,-1,-1,-1,2,1,-1,-1,1,1,1,-1,-1,0,1,1,1,1,1,1,-1,0,0,0,-1,-1,-1,-1,-1,1,-1,2

%N Dirichlet inverse of A327936, which is multiplicative sequence with a(p^e) = p if e >= p, otherwise 1.

%C Multiplicative because A327936 is.

%H Antti Karttunen, <a href="/A359432/b359432.txt">Table of n, a(n) for n = 1..65537</a>

%F Multiplicative with a(p^e) = (1 - p)^(e/p) if p | e, -(1 - p)^((e - 1)/p) if e == 1 (mod p), and 0 otherwise. - _Amiram Eldar_, Jan 26 2023

%t f[p_, e_] := Switch[Mod[e, p], 0, (1 - p)^(e/p), 1, -(1 - p)^((e - 1)/p), _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}] (* _Amiram Eldar_, Jan 26 2023 *)

%o (PARI)

%o A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };

%o memoA359432 = Map();

%o A359432(n) = if(1==n,1,my(v); if(mapisdefined(memoA359432,n,&v), v, v = -sumdiv(n,d,if(d<n,A327936(n/d)*A359432(d),0)); mapput(memoA359432,n,v); (v)));

%Y Cf. A327936.

%Y Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).

%Y Cf. also A358216, A359433.

%K sign,mult

%O 1,27

%A _Antti Karttunen_, Jan 02 2023